The take away from that video is the producer has an agenda and is ill-informed. Fat from healthy animals is very healthy for the human body. Refined carbohydrates are 100 times more likely to cause the issues mentioned above. I don't want to create too many straw men here but the general idea is that humans lived on fat and meat and whatever roots and berries they could find for quite a long time.
Fat is necessary for proper hormone function, brain function, muscle function, pretty much anything that needs to be done in the body. Some argue that we don't need to ingest fat since we can make it, but the same could be said for carbs, which can easily be created in the liver through gluconeogenisis from both fats and proteins. The only thing we need carbs in the blood for (in the form of glucose) is brain function. Everything else can run off ketones from fat.
There's a lot more to be said, but in general, carbs, protein, and fats are not bad in and of themselves. Some are worse than others, and different people respond differently to them. However, fat is not dangerous unless we begin to use the fake oils, just like the refined carbs promote a lot of negative changes.